On some special classes of contact B0B_0-VPG graphs

A graph $G$ is a $B_0$-VPG graph if one can associate a path on a rectangular grid with each vertex such that two vertices are adjacent if and only if the corresponding paths intersect at at least one grid-point. A graph $G$ is a contact $B_0$-VPG graph if it is a $B_0$-VPG graph admitting a representation with no two paths crossing and no two paths sharing an edge of the grid. In this paper, we present a minimal forbidden induced subgraph characterisation of contact $B_0$-VPG graphs within four special graph classes: chordal graphs, tree-cographs, $P_4$-tidy graphs and $P_5$-free graphs. Moreover, we present a polynomial-time algorithm for recognising chordal contact $B_0$-VPG graphs.Comment: 34 pages, 15 figure

The structure of imperfect critically strongly-imperfect graphs

AbstractThe family of all critically strongly-imperfect graphs decomposes in two nonempty classes: perfect and imperfect ones. In this paper we characterize the critically strongly-imperfect graphs which are, simultaneously, imperfect. We prove that these are precisely the holes of odd length ⩾ 5 or their complements

An integer analogue of Carathéodory's theorem

AbstractWe prove a theorem on Hilbert bases analogous to Carathéodory's theorem for convex cones. The result is used to give an upper bound on the number of nonzero variables needed in optimal solutions to integer programs associated with totally dual integral systems. For integer programs arising from perfect graphs the general bounds are improved to show that if G is a perfect graph with n nodes and w is a vector of integral node weights, then there exists a minimum w-covering of the nodes that uses at most n distinct cliques